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Random regression models (RRM) have been applied successfully to the analysis of test day production records of dairy cattle in Canada. RRM can be applied to any trait that is observed on an animal over time. Survival is a trait that is implicitly observed many times during the life of any animal. Veerkamp, Brotherstone, and Meuwissen (1999) suggested the use of RRM for survival analyses. This proposal is based on their suggestion, but is not exactly the same model as theirs.
At any given point in time an animal is either alive or dead (1 or 0). The survival 'curve' of an individual is a straight line with the value of 1 from birth or first calving up until it is culled whereupon the value of 0 is assigned from that point until the maximum age limit is reached. The survival 'curve' of a population of cows represents the average survival rate at various points in time, but it starts at 1 and decreases to zero (an age when all animals would be dead).
Survival 'curves' are expected to differ depending on the production level of the cow and depending on the type classification of the cow. The year and season of birth could also have an influence on the shape of the survival curve, and lastly, herd environment and management could have an effect on survival.
The objective of this report is to propose a RR model for the analysis of survival data. A small, simplified example is given.
The necessary items needed for each cow are Cow ID, Herd ID, Year-Season of Birth, Birthdate, Date Culled, Age When Culled, Latest Production EBVs, and Latest Conformation Scores. Reasons for culling may also be necessary at some point.
If Date Culled is blank, then the cow is still active. From this data an observation vector, , can be constructed for the cow. If ages 28 to 100 are considered in the analysis, for example, then is a vector of length 73. If the cow was culled at age 40, then the first 12 elements of would be 1 and the remaining 61 elements would be zero. If a cow was not yet culled and was 40 months of age, then would be of length 13 only and all elements equal to 1. Every animal would have a large number of 'observations'.
Production EBVs would be assigned to one of five levels depending upon the year of birth of the cow, so that roughly equal numbers of cows are in each level. The assumption is that survival depends on the genetic level of production. Similarly, conformation traits would be assigned to one of five levels by year of birth. The 1 to 9 categories could be condensed to five. Survival depends on the classifications of the cow. Particular traits may have more influence than others.
An observation, 0 or 1, is observed on an animal at age iborn in year-season j, and herd k within year-season j.
represent the vector of at values for all animals,
represent the vector of pt values for all animals that
were observed, and
represent the vector of residual effects,
Note that at age 28 the majority of animals (if not all) will have an observation of 1, which means that the phenotypic variance would be zero or very close to it. As animals age, more observations become 0, and the phenotypic variance increases until there are an equal number of 0's and 1's. After this point, there are more 0's and fewer 1's so that the phenotypic variance decreases until it reaches 0, at which point all animals have been culled. If first lactations begin mostly at 18 to 24 months of age, then 28 months would be near the end of the first lactation and the phenotypic variance should not be 0.
In order to simplify the illustration of the methods, let ages be exchanged with lactation numbers (1 to 5 only). Below are the data vectors for 16 animals. In total there are 80(=516) observations. Assumptions are that these 16 animals were all born in the same YS in one herd and belong to the same levels of production and conformation.
The simplified model is
The solutions from the mixed model equations were
There are several ways to convert these numbers into
useful EBVs. One way is to determine differences
among animals at a fixed age, such as lactation number
five. Then xi=1, and
This LaTeX document is available as postscript or asAdobe PDF.Larry Schaeffer